simpeg.regularization.SmoothnessFirstOrder.f_m_deriv#

SmoothnessFirstOrder.f_m_deriv(m)[source]#

Derivative of the regularization kernel function.

For first-order smoothness regularization in the x-direction, the derivative of the regularization kernel function with respect to the model is given by:

$$\frac{\partial {\mathbf{f}}_{\mathbf{m}}}{\partial \mathbf{m}}={\mathbf{G}}_{\mathbf{x}}$$

where ${\mathbf{G}}_{\mathbf{x}}$ is the partial cell gradient operator along x (i.e. the x-derivative).

Parameters:

mnumpy.ndarray

The model.

Returns:

scipy.sparse.csr_matrix

The derivative of the regularization kernel function.

Notes

The objective function for first-order smoothness regularization along the x-direction is given by:

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}{\mathbf{G}}_{\mathbf{x}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]{\Vert }^2$$

where $\mathbf{m}$ are the discrete model parameters (model), ${\mathbf{m}}^{(ref)}$ is the reference model, ${\mathbf{G}}_{\mathbf{x}}$ is the partial cell gradient operator along the x-direction (i.e. x-derivative), and $\mathbf{W}$ is the weighting matrix. Similar for smoothness along y and z. See the SmoothnessFirstOrder class documentation for more detail.

We define the regularization kernel function ${\mathbf{f}}_{\mathbf{m}}$ as:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})={\mathbf{G}}_{\mathbf{x}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]$$

such that

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}{\mathbf{f}}_{\mathbf{m}}{\Vert }^2$$

The derivative with respect to the model is therefore:

$$\frac{\partial {\mathbf{f}}_{\mathbf{m}}}{\partial \mathbf{m}}={\mathbf{G}}_{\mathbf{x}}$$